LAMPLIGHTER GROUPS AND VON NEUMANN‘S CONTINUOUS REGULAR RING
GABOR´ ELEK
Abstract. Let Γ be a discrete group. Following Linnell and Schick one can define a continuous ring c(Γ) associated with Γ. They proved that if the Atiyah Conjecture holds for a torsion-free group Γ, then c(Γ) is a skew field. Also, if Γ has torsion and the Strong Atiyah Conjecture holds for Γ, then c(Γ) is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group Γ = Z2 ≀ Z. It is known that C(Z2 ≀ Z) does not even have a classical ring of quotients. Our main result is that if H is amenable, then c(Z2 ≀ H) is isomorphic to a continuous ring constructed by John von Neumann in the 1930′s. Keywords. continuous rings, von Neumann algebras, the algebra of affiliated operators, lamplighter group
1. Introduction
Let us consider Matk×k(C) the algebra of k by k matrices over the complex field. This ring is a unital ∗-algebra with respect to the conjugate transposes. For each element ∗ A ∈ Matk×k(C) one can define A satisfying the following properties. • (λA)∗ = λA∗ • (A + B)∗ = A∗ + B∗ • (AB)∗ = B∗A∗ • 0∗ =0, 1∗ =1 Also, each element has a normalized rank rk(A) = Rank(A)/k with the following properties. • rk(0) = 0, rk(1) = 1, • rk(A + B) ≤ rk(A)+rk(B) • rk(AB) ≤ min{rk(A), rk(B)} • rk(A∗) = rk(A) • If e and f are orthogonal idempotents then rk(e + f) = rk(e)+rk(f).
The ring Matk×k(C) has an algebraic property namely, von Neumann called regularity: Any principal left-(or right) ideal can be generated by an idempotent. Furthermore, among these generating idempotents there is a unique projection (that is Matk×k(C)isa ∗-regular ring). In a von Neumann regular ring any non-zerodivisor is necessarily invertible. One n ∗ can also observe that the algebra of matrices is proper, that is i=1 aiai = 0 implies that all the matrices ai are zero. One should note that if R is a ∗-regular ring with a rank P AMS SUBJECT CLASSIFICATION: 16A30, 46L10 RESEARCH PARTLY SPONSORED BY MTA RENYI “LENDULET” GROUPS AND GRAPHS RESEARCH GROUP 1 2 GABOR´ ELEK function, then the rank extends to Matk×k(R) [6], where the extended rank has the same property as rk except that the rank of the identity is k. One can immediately see that the rank function defines a metric d(A, B) := rk(A − B) on any algebra with a rank, and the matrix algebra is complete with respect to this metric. These complete ∗-regular algebras are called continuous ∗-algebras (see [5] for an extensive study of continuous rings). Note that for the matrix algebras the possible values of the rank functions are 0, 1/k, 2/k,..., 1. John von Neumann observed that there are some interesting examples of infinite dimensional continuous ∗-algebras, where the rank function can take any real values in between 0 and 1. His first example was purely algebraic. Example 1. Let us consider the following sequence of diagonal embeddings.
C → Mat2×2(C) → Mat4×4(C) → Mat8×8(C) → ...
One can observe that all the embeddings are preserving the rank and the ∗-operation. Hence the direct limit lim Mat k k (C) is a ∗-regular ring with a proper rank function. −→ 2 ×2 The addition, multiplication, the ∗-operation and the rank function can be extended to the metric completion M of the direct limit ring. The resulting algebra M is a simple, proper, continuous ∗-algebra, where the rank function can take all the values on the unit interval.
Example 2. Consider a finite, tracial von Neumann algebra N with trace function trN . Then N is a ∗-algebra equipped with a rank function. If P is a projection, then rkN (P )= t ∞ trN (P ). For a general element A ∈N , rkN (A)=1−limt→∞ 0 trN (Eλ)dλ, where 0 Eλ dλ is the spectral decomposition of A∗A. In general, N is not regular, but it has the Ore property with respect to its zero divisors. The Ore localizaRtion of N with respectR to its non-zerodivisors is called the algebra of affiliated operators and denoted by U(N ). These algebras are also proper continuous ∗-algebras [1]. The rank of an element A ∈ U(N ) is given by the trace of the projection generating the principal ideal U(N )A. It is important to note, that U(N ) is the rank completion of N (Lemma 2.2 ([12]). Linnell and Schick observed [9] that if X is a subset of a proper ∗-regular algebra R, then there exists a smallest ∗-regular subalgebra containing X, the ∗-regular closure. Now let Γ be a countable group and CΓ be its complex group algebra. Then one can consider the natural embedding of the group algebra to its group von Neumann algebra CΓ →N Γ. Let U(Γ) denote the Ore localization of N (Γ) and the embedding CΓ → U(Γ). Since U(Γ) is a proper ∗-regular ring, one can consider the smallest ∗-algebra A(Γ) in U(Γ) containing C(Γ). Let c(Γ) be the completion of the algebra A above. It is a continuous ∗-algebra [5]. Of course, if the rank function has only finitely many values in A, then c(Γ) equals to A(Γ). Note that if CΓ is embedded into a continuous ∗-algebra T , then one can still define cT (Γ) as the smallest continuous ring containing CΓ. In [3] we proved that if Γ is amenable, c(Γ) = cT (Γ) for any embedding CΓ → T associated to sofic representations of Γ, hence c(Γ) can be viewed as a canonical object. Linnell and Schick calculated the algebra c(Γ) for several groups, where the rank function has only finitely many values on A. They proved the following results: LAMPLIGHTER GROUPS AND VON NEUMANN‘S CONTINUOUS REGULAR RING1 3
• If Γ is torsion-free and the Atiyah Conjecture holds for Γ, then c(Γ) is a skew-field. This is the case, when Γ is amenable and CΓ is a domain. Then c(Γ) is the Ore localization of CΓ. If Γ is the free group of k generators, then c(Γ) is the Cohen- Amitsur free skew field of k generators. The Atiyah Conjecture for a torsion-free group means that the rank of an element in Matk×k(CΓ) ⊂ Matk×k(U(N (Γ))) is an integer. • If the orders of the finite subgroups of Γ are bounded and the Strong Atiyah Con- jecture holds for Γ, then c(Γ) is a finite dimensional matrix ring over some skew field. In this case the Strong Atiyah Conjecture means that the ranks of an element 1 in Matk×k(CΓ) ⊂ Matk×k(U(N (Γ))) is in the abelian group Z, where lcm(Γ) lcm(Γ) indicates the least common multiple of the orders of the finite subgroups of Γ.
The lamplighter group Γ = Z2 ≀ Z has finite subgroups of arbitrarily large orders. Also, although Γ is amenable, CΓ does not satisfy the Ore condition with respect to its non- zerodivisors [8]. In other words, it has no classical ring of quotients. The goal of this paper is to calculate c(Z2 ≀Z) and even c(Z2 ≀H), where H is a countably infinite amenable group.
Theorem 1. If H is a countably infinite amenable group, then c(Z2 ≀ H) is the simple continuous ring M of von Neumann.
2. Crossed Product Algebras In this section we recall the notion of crossed product algebras and the group-measure space construction of Murray and von Neumann. Let A be a unital, commutative ∗-algebra and φ : Γ → Aut(A) be a representation of the countable group Γ by ∗-automorphisms. The associated crossed product algebra A ⋊ Γ is defined the following way. The elements of A ⋊ Γ are the finite formal sums
aγ · γ, γ∈Γ X
where aγ ∈A. The multiplicative structure is given by
δ · aγ = φ(δ)(aγ) · δ. The ∗-structure is defined by γ∗ = γ−1 and (γ · a)∗ = a∗ · γ−1. Note that
∗ ∗ ∗ ∗ −1 ∗ −1 ∗ ∗ (δ · aγ) =(φ(δ)aγ · δ) = δ · φ(δ)aγ = φ(δ )φ(δ)aγ · δ = aγ · δ . Now let (X,µ) be a probability measure space and τ : Γ y X be a measure preserv- ing action of a countable group Γ on X. Then we have a ∗-representationτ ˆ of Γ in Aut(L∞(X,µ)), where L∞(X,µ) is the commutative ∗-algebra of bounded measurable functions on X (modulo zero measure perturbations).
τˆ(γ)(f)(x)= f(τ(γ−1)(x)) . 4 GABOR´ ELEK
Let H = l2(Γ,L2(X,µ)) be the Hilbert-space of L2(X,µ)-valued functions on Γ. That is, each element of H can be written in the form of
bγ · γ, γ∈Γ X 2 ∞ ⋊ where γ∈Γ kbγk < ∞ . Then we have a representation L of L (X,µ)) Γ on l2(Γ,L2(X,µ)) by P
L( aγ · γ)( bδ · δ)= aγ(ˆτ(γ)(βδ)) · γδ . γ∈Γ γ∈Γ ! X Xδ∈Γ Xδ∈Γ X ∞ ⋊ Note that L( γ∈Γ aγ · γ) is always a bounded operator. A trace is given on L (X,µ)) Γ by P Tr(S)= a1(x)dµ(x) . ZX ∞ ⋊ 2 2 The weak operator closure of L(Lc (X,µ)) Γ) in B (l (Γ,L (X,µ))) is the von Neumann ∞ algebra N (τ) associated to the action. Here Lc (X,µ) denotes the subspace of functions in L∞(X,µ) having only countable many values. Note that one can extend Tr to TrN (τ) on the von Neumann algebra to make it a tracial von Neumann algebra. ∞ ⋊ We will denote by c(τ) the smallest continuous algebra in U(N (τ)) containing Lc (X,µ) Γ. ∞ ⋊ 2 2 One should note that the weak closure of Lc (X,µ) Γ in B (l (Γ,L (X,µ))) is the same as the weak closure of L∞(X,µ) ⋊ Γ. Hence our definition for the von Neumann algebra ∞ ⋊ of an action coincides with the classical definition. On the other hand, c(Lc (X,µ) Γ) is smaller than c(L∞(X,µ) ⋊ Γ).
3. The Bernoulli Algebra
Let H be a countable group. Consider the Bernoulli shift space BH := h∈H {0, 1} with the usual product measure νH . The probability measure preserving action τH : H y Q (BH ,νH ) is defined by −1 τH (δ)(x)(h)= x(δ h) , where x ∈ BH , δ, h ∈ H. Let AH be the commutative ∗-algebra of functions that depend only on finitely many coordinates of the shift space. It is well-known that the Rademacher functions {RS}S⊂H, |S|<∞ form a basis in AH , where
RS(x)= exp(iπx(δ)) . δY∈S The Rademacher functions with respect to the pointwise multiplication form an Abelian group isomorphic to ⊕h∈H Z2 the Pontrjagin dual of the compact group BH satisfying
• RSRS′ = RS△S′
• BH RS dν =0, if |S| > 0 • R = 1. R ∅ LAMPLIGHTER GROUPS AND VON NEUMANN‘S CONTINUOUS REGULAR RING2 5
The group H acts on AH by −1 τˆH (δ)(f)(x)= f τH (δ )(x) . Hence,
τˆH (δ)RS = RδS .
Therefore, the elements of AH ⋊ H can be uniquely written as in the form of the finite sums
cδ,SRS · δ, S Xδ X where δ · RS = RδS · δ. Now let us turn our attention to the group algebra C(Z2 ≀ H). For δ ∈ H, let tδ be the C Z generator in h∈H Z2 belonging to the δ-component. Any element of ( 2 ≀ H) can be written in a unique way as a finite sum P cδ,StS · δ, S Xδ X ′ ′ where tS = s∈S ts, δ · tS = tδS, tStS = tS△S . Also note that
Q Tr( cδ,StS · δ)= c1,∅ . S Xδ X Hence we have the following proposition. Proposition 3.1. There exists a trace preserving ∗-isomorphism κ : C(Z2 ≀ H) →AH ⋊ H such that
κ( cδ,StS · δ)= cδ,SRS · δ. S S Xδ X Xδ X Recall that if A ⊂ N1, B ⊂ N1 are weakly dense *-subalgebras in finite tracial von Neumann algebras N1 and N2 and κ : A → B is a trace preserving ∗-homomorphism, then κ extends to a trace preserving isomorphism between the von Neumann algebras themselves (see e.g. [7] Corollary 7.1.9.). Therefore, κ : C(Z2 ≀ H) → AH ⋊ H extends to a trace (and hence rank) preserving isomorphism between the von Neumann algebras N (Z2 ≀ H) and N (τH ). Proposition 3.2. For any countable group H, ∼ c(Z2 ≀ H) = c(τH ) .
Proof. The rank preserving isomorphism κ : N (Z2 ≀H) →N (τH ) extends to a rank preserv- ing isomorphism between the rank completions, that is, the algebras of affiliated operators. ⋊ ∞ ⋊ It is enough to prove that the rank closure of AH H is Lc (BH ,νH ) H. ∞ Lemma 3.1. Let f ∈ Lc (BH ,νH ). Then rkN (τH )(f)= νH (supp(f)). 6 GABOR´ ELEK
Proof. By definition,
rkN (τH )(f)=1 − lim trN (τH )Eλ , λ→0 ∗ where Eλ is the spectral projection of f f corresponding to λ.
2 trN (τH )Eλ = νH ({x | |f (x)|≤ λ}) . 2 Hence, rkN (τH )(f)=1 − νH ({x | f (x)=0})= νH (supp(f)) . ∞ rk ∞ rk Let {mn}n=1 ⊂ AH ,mn → m ∈ Lc (BH ,νH ). Then mn · γ → m · γ. Therefore our proposition follows from the lemma below.
∞ Lemma 3.2. AH is dense in Lc (BH ,νH ) with respect to the rank metric. ∞ ∞ ∞ Proof. By Lemma 3.1, Lfin(BH ,νH ) is dense in Lc (BH ,νH ), where Lfin(BH ,νH ) is the ∗-algebra of functions taking only finitely many values. Recall that V ⊂ BH is a basic set if 1V ∈AH . It is well-known that any measurable set in BH can be approximated by basic ∞ sets, that is for any U ⊂ BH , there exists a sequence of basic sets {Vn}n=1 such that
(1) lim νH (Vn△U)=0 . n→∞ By (1) and Lemma 3.1
lim rkN (τn)(1Vn − 1U )=0 . n→∞ l m Let f = m=1 cm1Um , where Um are disjoint measurable sets. Let limn→∞ νH (Vn △Um)= 0, where {V m}∞ are basic sets. Then P n n=1 l
m lim rkN (τn)( cm1Vn − f)=0 . n→∞ m=1 X ∞ Therefore, AH is dense in Lfin(BH ,νH ) .
4. The Odometer Algebra The Odometer Algebra is constructed via the odometer action using the algebraic crossed product construction. Let us consider the compact group of 2-adic integers Zˆ (2). Recall that Zˆ (2) is the completion of the integers with respect to the dyadic metric −k d(2)(n,m)=2 , where k is the power of two in the prime factor decomposition of |m − n|. The group Zˆ (2) can be identified with the compact group of one way infinite sequences with respect to the binary addition. Zˆ l n n The Haar-measure µhaar on (2) is defined by µhaar(Un)=1/2 , where 0 ≤ l ≤ 2 − 1 l Zˆ n and Un is the clopen subset of elements in (2) having residue l modulo 2 . Let T be the Zˆ Z y Zˆ addition map x → x +1 in (2). The map T defines an action ρ : ( (2),µhaar) Zˆ The dynamical system (T, (2),µhaar) is called the odometer action. As in Section 3, we LAMPLIGHTER GROUPS AND VON NEUMANN‘S CONTINUOUS REGULAR RING3 7
∞ Zˆ consider the ∗-subalgebra of function AM in L ( (2),µhaar) that depend only on finitely n many coordinates of Zˆ (2). We consider a basis for AM . For n ≥ 0 and 0 ≤ l ≤ 2 − 1 let 2πix(mod 2n) F l (x)= exp l . n 2n 2l l l Notice that Fn+1 = Fn. Then the functions {Fn}n,l|(l,n)=1 form the Pr¨ufer 2-group
Z(2) = Z1 ⊂ Z2 ⊂ Z4 ⊂ Z8 ⊂ ... with respect to the pointwise multiplication. The discrete group Z(2) is the Pontrjagin dual Zˆ 1 of the compact Abelian group (2). The element Fn is the generator of the cyclic subgroup Z2n . Note that l Fn dµ =0 Zˆ haar Z (2) l Z except if l =0,n = 0, when Fn ≡ 1. Observe that if k ∈ then l l+k(mod 2n) (2) ρ(k)Fn = Fn n l l+k(mod 2 ) since Fn(x − k)= Fn (x) . Hence we have the following lemma.
Lemma 4.1. The elements of AM ⋊Z can be uniquely written as finite sums in the form l cn,l,kFn · k, n≥0 Xk X l|(Xl,n)=1 n l l+k(mod 2 ) 0 where k · Fn = Fn and F0 =1.
5. Periodic operators Definition 5.1. A function Z × Z → C is a periodic operator if there exists some n ≥ 1 such that • A(x,y)=0, if |x − y| > 2n • A(x,y)= A(x +2n,y +2n). Observe that the periodic operators form a ∗-algebra, where • (A + B)(x,y)= A(x,y)+ B(x,y)
• AB(x,y)= z∈Z A(x,z)B(z,y) • A∗(x,y)= A(y,x) P Proposition 5.1. The algebra of periodic operators P is ∗-isomorphic to a dense subalgebra of M. Proof. We call A ∈P an element of type-n if • A(x,y)= A(x +2n,y +2n) • A(x,y)=0if0 ≤ x ≤ 2n − 1, y > 2n − 1 • A(x,y)=0if0 ≤ x ≤ 2n − 1, y < 0. 8 GABOR´ ELEK
Clearly, the elements of type-n form an algebra Pn isomorphic to Mat2n×2n (C) and Pn → Pn+1 is the diagonal embedding. Hence, we can identify the algebra of finite type elements P = ∪∞ P with lim Mat n n (C). f n=1 n −→ 2 ×2 For A ∈P, if n ≥ 1 is large enough, let An ∈Pn be defined the following way. n n n • An(x,y)= A(x,y) if 2 l ≤ x,y ≤ 2 l +2 − 1 for some l ∈ Z. • Otherwise, A(x,y)=0.
∞ Lemma5.1. (i): {An}n=1 is a Cauchy-sequence in M. (ii): (A + B)n = An + Bn. ∗ ∗ (iii): rkM(An − (A )n)=0. (iv): rkM((ABn) − AnBn)=0,. (v): limn→∞ An =0 if and only if A =0.
Proof. First observe that for any Q ∈Pn |{0 ≤ x ≤ 2n − 1 |∃ 0 ≤ y ≤ 2n − 1 such that A (x,y) =6 0.}| rk (Q) ≤ n M 2n Suppose that A(x,y)= A(x +2k,y +2k) and k
n l Proof. For n ≥ 1 and 0 ≤ l ≤ 2 − 1 let En ∈D be defined by 2πix(mod 2n) El (x,x) := exp l . n 2n 2l l 1 It is easy to see that En+1 = En and the multiplicative group generated by En is isomorphic Z n l to 2 . Observe that the set {En}n,l,(l,n)=1 form a basis in the space of n-type diagonal ∼ ∞ C n C Z operators. Therefore, D = ∪n=1 (Z2 )= ( (2)). Let J ∈P be the following element. • J(x,y)=1, if y = x + 1. • Otherwise, J(x,y)=0. Then l l+1(mod 2n) (3) J · En = En . LAMPLIGHTER GROUPS AND VON NEUMANN‘S CONTINUOUS REGULAR RING4 9
Also, any periodic operator A can be written in a unique way as a finite sum k Dk · J , Z Xk∈ where Dk is a diagonal operator in the form ∞ l Dk = cl,n,kEn . n=0 X l|(Xl,n)=1 Thus, by (2) and (3), we have the following corollary.
Corollary 5.1. The map ψ : P→AM ⋊Z defined by l l ψ( cl,n,kEn · k)= cl,n,kFn · k n≥0 n≥0 Xk X l|(Xl,n)=1 Xk X l|(Xl,n)=1 is a ∗-isomorphism of algebras.
6. Luck’s¨ Approximation Theorem revisited The goal of this section is to prove the following proposition. Proposition 6.1. We have c(ρ) ∼= M where ρ is the odometer action. Proof. Let us define the linear map t : P → C by n 2 −1 A(i,i) t(A) := i=0 , 2n P where A ∈P and A(x +2n,y +2n) for all x,y ∈ Z.
Lemma 6.1. TrN (ρ)(ψ(A)) = t(A) , where ψ is the ∗-isomorphism of Corollary 5.1. l l Proof. Recall that TrN (ρ)(Fn) = 0, except, when l =0,n =0,Fn =1. If n =6 0 and l =6 0, l l then t(En) is the sum of all k-th roots of unity for a certain k, hence t(En) = 0. Also, t(1) = 1. Thus, the lemma follows.
It is enough to prove that
(4) rkM(A)=rkN (ρ)(ψ(A))
Indeed by (4), ψ is a rank-preserving ∗-isomorphism between P and AM ⋊Z. Hence the isomorphism ψ extends to a metric isomorphism
ψˆ : P → AM ⋊Z ,
where P is the closure of P in M and AM ⋊Z is the closure of AM ⋊Z in U(N (ρ)) . ∼ Since P is dense in M, P = M. Also, AM ⋊Z is a ∗-subalgebra of U(N (ρ)), since the ∗-ring operations are continuous with respect to the rank metric. Therefore AM ⋊Z is a continuous algebra isomorphic to M. Observe that the rank closure AM ⋊Z is isomorphic ∞ Zˆ ⋊Z to the rank closure of Lc ( (2),µhaar) by the argument of Lemma 3.2. Therefore, c(ρ) ∼= M. Thus from now on, our only goal is to prove (4). 10 GABOR´ ELEK
Lemma 6.2. Let A ∈ P and An ∈ Mat2n×2n (C) as in Section 5. Then the matrices ∞ {An}n=1 have uniformly bounded norms. Proof. Let M,N be chosen in such a way that
• |An(x,y)|≤ M for any x,y ∈ Z,n ≥ 1. N • |An(x,y)| =0 if |x − y|≥ 2 . n Now let v =(v(1),v(2),...,v(2n)) ∈ C2 , kvk2 =1 . Then 2n 2n 2 2 2 2 kAnvk = | An(x,y)v(y)| ≤ M | v(y)| ≤ x=1 x=1 X y ||x−Xy|
∞ or all f ∈ C[0,K]. As in [10], we can see that the measures {µn}n=1 converge weakly to µ. Indeed by Lemma 6.3, ∗ ∗ lim t(P (AnAn)) = TrN (ρ)P (A A) n→∞ for any real polynomial P , therefore ∗ ∗ lim t(f(AnAn)) = TrN (ρ)f(A A) n→∞ for all f ∈ C[0,K]. ∗ ∗ Since rkM(An)=rkM(AnAn) and rkN (ρ)(ψ(A)) = rkN (ρ))ψ(A A)), in order to prove (4) it is enough to see that ∗ ∗ lim rkM(AnAn)=rkN (ρ)(ψ(A A)) . n→∞ ∗ Observe that rkM(AnAn)=1 − µn(0) and ∗ rkN (ρ)(ψ(A A))=1 − lim TrN (ρ)Eλ = µ(0) . λ→0 Hence, our proposition follows from the lemma below (an analogue of L¨uck’s Approximation Theorem).
Lemma 6.4. limn→∞ µn(0) = µ(0) .
λ λ Proof. Let Fn(λ)= 0 µn(t) dt and F (λ)= 0 µ(t) dt be the distribution functions of our ∞ spectral measures. Since {µn}n=1 weakly converges to the measure µ, it is enough to show ∞ R R n n n C m m C that {Fn}n=1 converges uniformly. Let n ≤ m and Dm : Mat2 ×2 ( ) → Mat2 ×2 ( ) be the diagonal operator. Let ε> 0. By Lemma 5.1, if n,m are large enough, n m Rank(Dm(An) − Am) ≤ ε2 . Hence, by Lemma 3.5 [2],
kFn − Fmk∞ ≤ ε. ∞ Therefore, {Fn}n=1 converges uniformly.
7. Orbit Equivalence
First let us recall the notion of orbit equivalence. Let τ1 : Γ1 y (X,µ) resp. τ2 : Γ2 y (Y,ν) be essentially free probability measure preserving actions of the countably infinite groups Γ1 resp. Γ2. The two actions are called orbit equivalent if there exists a measure preserving bijection Ψ : (X,µ) → (Y,ν) such that for almost all x ∈ X and γ ∈ Γ1 there exists γx ∈ Γ2 such that
τ2(γx)(Ψ(x)) = Ψ(τ1(γ)(x)) . ∼ Feldman and Moore [4] proved that if τ1 and τ2 are orbit equivalent then N (τ1) = N (τ2) . The goal of this section is to prove the following proposition. ∼ Proposition 7.1. If τ1 and τ2 are orbit equivalent actions, then c(τ1) = c(τ2). 12 GABOR´ ELEK
Our Theorem 1 follows from the proposition. Indeed, by Proposition 3.2 and Proposition 6.1 ∼ ∼ M = c(ρ) and c(Z2 ≀ H) = c(τH ) . By the famous theorem of Ornstein and Weiss [11], the odometer action and the Bernoulli shift action of a countably infinite amenable group are orbit ∼ equivalent. Hence M = c(Z2 ≀ H) . Proof. We build the proof of our proposition on the original proof of Feldman and Moore. Let γ ∈ Γ1, δ ∈ Γ2. Let −1 M(δ, γ)= {y ∈ Y | τ2(δ)(y)=Ψ(τ1(γ)Ψ (y))}
−1 N(γ,δ)= {x ∈ X | τ1(γ)(x)=Ψ (τ2(δ)Ψ(x))}. Observe that Ψ(N(δ, γ)) = M(γ,δ) . Following Feldman and Moore ([4], Proposition 2.1) for any γ ∈ Γ1, δ ∈ Γ2
κ(γ)= h · 1M(h,γ) hX∈Γ2 and
λ(δ)= g · 1N(g,δ) g∈Γ X1 k are well-defined. That is, n=1 hn · 1M(hn,γ) converges weakly to κ(γ) ∈N (τ2) as k → ∞ k ∞ and n=1 gn · 1N(gn,δ) converges weakly to λ(δ) ∈ N (τ1) as k → ∞, where {γn}n=1 resp. ∞ P {δn}n=1 are enumerations of the elements of Γ1 resp. Γ2. Furthermore,P one can extend κ resp. λ to maps ′ ∞ κ : L ((X,µ) ⋊ Γ1) →N (τ2) resp. ′ ∞ λ : L ((Y,ν) ⋊ Γ2) →N (τ1) by ∞ ′ −1 −1 κ ( aγ · γ)= (aγ ◦ Ψ ) · κ(γ)= (aγ ◦ Ψ ) · hn · 1M(hn,γ) γ∈Γ γ∈Γ γ∈Γ n=1 X1 X1 X1 X and ∞ ′ λ ( bδ · δ)= (bδ ◦ Ψ) · λ(δ)= (bδ ◦ Ψ) · gn · 1N(gn,δ) . n=1 δX∈Γ2 δX∈Γ2 δX∈Γ2 X The maps κ′ resp. λ′ are injective trace-preserving ∗-homomorphisms with weakly dense ranges. Hence they extend to isomorphisms of von Neumann algebras
κˆ : N (τ1) →N (τ2), λˆ : N (τ2) →N (τ1) , whereκ ˆ and λˆ are, in fact, the inverses of each other. LAMPLIGHTER GROUPS AND VON NEUMANN‘S CONTINUOUS REGULAR RING6 13
Lemma 7.1. k −1 (5) lim rkN (τ2) (aγ ◦ Ψ ) · hn · 1M(hn,γ) − κˆ( aγ · γ) =0 . k→∞ γ∈Γ n=1 γ∈Γ ! X1 X X1
k ˆ (6) lim rkN (τ1) (bδ ◦ Ψ) · gn · 1N(gn,δ) − λ( bδ · δ) =0 . k→∞ n=1 ! δX∈Γ2 X δX∈Γ2 ∞ Proof. By definition, the disjoint union ∪n=1M(hn,γ) equals to Y (modulo a set of measure k ∞ zero). We need to show that if { n=1 Tn · 1M(hn,γ)}k=1 weakly converges to an element k ∞ S ∈N (τ ), then { T · 1 n } converges to S in the rank metric as well, where 2 n=1 n M(h ,γP) k=1 ∞ ⋊ k 2 2 ˆ Tn ∈ Lc (Y,ν) Γ2. Let Pk = n=1 1M(hn,γ) ∈ l (Γ,L (Y,ν)). We denote by Pk the k P ∞ element 1 n in L (Y,ν) ⋊ Γ . By definition, if L(A)(P ) = 0 then APˆ = 0. n=1 M(h ,γ) c P 2 k k Now, by weak convergence, P l
L(S)(Pk)= lim Tn · 1M(hn,γ)(Pk) . l→∞ n=1 X That is, k
L(S − Tn · 1M(hn,γ))(Pk)=0 . n=1 X Therefore, k ˆ (S − Tn · 1M(hn,γ))Pk =0 . n=1 X Thus, k k ˆ (S − Tn · 1M(hn,γ))=(S − Tn · 1M(hn,γ))(1 − Pk) . n=1 n=1 X X ˆ k By Lemma 3.1, rkN (τ2)(1 − Pk)=1 − n=1 ν(M(hn,γ)), hence Pk lim rkN (τ2)(S − Tn · 1M(hn,γ))=0 . k→∞ n=1 X ∞ ⋊ Now let us turn back to the proof of our proposition. By (5),κ ˆ maps the algebra Lc (X,µ) ∞ ⋊ Γ1 into the rank closure of Lc (Y,ν) Γ2. Sinceκ ˆ preserves the rank,κ ˆ maps the rank ∞ ⋊ ∞ ⋊ ˆ closure of Lc (X,µ) Γ1 into the rank closure of Lc (Y,ν) Γ2. Similarly, λ maps the rank ∞ ⋊ ∞ ⋊ closure of Lc (Y,ν) Γ2 into the rank closure of Lc (X,µ) Γ1. That is,κ ˆ provides an ∞ ⋊ ∞ ⋊ isomorphism between the rank closures of Lc (X,µ) Γ1 and Lc (Y,ν) Γ2. Therefore, the ∞ ⋊ smallest continuous ring containing Lc (X,µ) Γ1 in U(N (τ1)) is mapped to the smallest ∞ ⋊ continuous ring containing Lc (Y,ν) Γ2 in U(N (τ2)) . 14 GABOR´ ELEK
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